Modern Control System Theory and Design, 2nd Edition

2.27.  TOTAL SOLUTION OF THE STATE EQUATION

The purpose of this section is to illustrate how one may obtain the complete solution for the output in the time domain of a control system utilizing the state-variable method. In this example, we will want to determine the complete solution by evaluating Eq. (2.255), the state transition equation.

Consider a system described by the following differential equation:

It is desired to determine the output c(t), given that the input r(t) is given by

and the initial conditions are c(0) = 1 and (0) = 0. The technique employed is to determine the state transition matrix from Eq. (2.256) and then evaluate Eq. (2.255) for x(t). The output c(t) is then evaluated from

If the state variables are defined by

and u(t) by

u(t) = r(t),

then the system can be described by the following two first-order differential equations:

Therefore, the system can be described ...